// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2011-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_CONJUGATE_GRADIENT_H
#define EIGEN_CONJUGATE_GRADIENT_H

namespace Eigen {

namespace internal {

    /** \internal Low-level conjugate gradient algorithm
  * \param mat The matrix A
  * \param rhs The right hand side vector b
  * \param x On input and initial solution, on output the computed solution.
  * \param precond A preconditioner being able to efficiently solve for an
  *                approximation of Ax=b (regardless of b)
  * \param iters On input the max number of iteration, on output the number of performed iterations.
  * \param tol_error On input the tolerance error, on output an estimation of the relative error.
  */
    template <typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
    EIGEN_DONT_INLINE void
    conjugate_gradient(const MatrixType& mat, const Rhs& rhs, Dest& x, const Preconditioner& precond, Index& iters, typename Dest::RealScalar& tol_error)
    {
        using std::abs;
        using std::sqrt;
        typedef typename Dest::RealScalar RealScalar;
        typedef typename Dest::Scalar Scalar;
        typedef Matrix<Scalar, Dynamic, 1> VectorType;

        RealScalar tol = tol_error;
        Index maxIters = iters;

        Index n = mat.cols();

        VectorType residual = rhs - mat * x;  //initial residual

        RealScalar rhsNorm2 = rhs.squaredNorm();
        if (rhsNorm2 == 0)
        {
            x.setZero();
            iters = 0;
            tol_error = 0;
            return;
        }
        const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
        RealScalar threshold = numext::maxi(RealScalar(tol * tol * rhsNorm2), considerAsZero);
        RealScalar residualNorm2 = residual.squaredNorm();
        if (residualNorm2 < threshold)
        {
            iters = 0;
            tol_error = sqrt(residualNorm2 / rhsNorm2);
            return;
        }

        VectorType p(n);
        p = precond.solve(residual);  // initial search direction

        VectorType z(n), tmp(n);
        RealScalar absNew = numext::real(residual.dot(p));  // the square of the absolute value of r scaled by invM
        Index i = 0;
        while (i < maxIters)
        {
            tmp.noalias() = mat * p;  // the bottleneck of the algorithm

            Scalar alpha = absNew / p.dot(tmp);  // the amount we travel on dir
            x += alpha * p;                      // update solution
            residual -= alpha * tmp;             // update residual

            residualNorm2 = residual.squaredNorm();
            if (residualNorm2 < threshold)
                break;

            z = precond.solve(residual);  // approximately solve for "A z = residual"

            RealScalar absOld = absNew;
            absNew = numext::real(residual.dot(z));  // update the absolute value of r
            RealScalar beta = absNew / absOld;       // calculate the Gram-Schmidt value used to create the new search direction
            p = z + beta * p;                        // update search direction
            i++;
        }
        tol_error = sqrt(residualNorm2 / rhsNorm2);
        iters = i;
    }

}  // namespace internal

template <typename _MatrixType, int _UpLo = Lower, typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar>> class ConjugateGradient;

namespace internal {

    template <typename _MatrixType, int _UpLo, typename _Preconditioner> struct traits<ConjugateGradient<_MatrixType, _UpLo, _Preconditioner>>
    {
        typedef _MatrixType MatrixType;
        typedef _Preconditioner Preconditioner;
    };

}  // namespace internal

/** \ingroup IterativeLinearSolvers_Module
  * \brief A conjugate gradient solver for sparse (or dense) self-adjoint problems
  *
  * This class allows to solve for A.x = b linear problems using an iterative conjugate gradient algorithm.
  * The matrix A must be selfadjoint. The matrix A and the vectors x and b can be either dense or sparse.
  *
  * \tparam _MatrixType the type of the matrix A, can be a dense or a sparse matrix.
  * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower,
  *               \c Upper, or \c Lower|Upper in which the full matrix entries will be considered.
  *               Default is \c Lower, best performance is \c Lower|Upper.
  * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
  *
  * \implsparsesolverconcept
  *
  * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
  * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
  * and NumTraits<Scalar>::epsilon() for the tolerance.
  * 
  * The tolerance corresponds to the relative residual error: |Ax-b|/|b|
  * 
  * \b Performance: Even though the default value of \c _UpLo is \c Lower, significantly higher performance is
  * achieved when using a complete matrix and \b Lower|Upper as the \a _UpLo template parameter. Moreover, in this
  * case multi-threading can be exploited if the user code is compiled with OpenMP enabled.
  * See \ref TopicMultiThreading for details.
  * 
  * This class can be used as the direct solver classes. Here is a typical usage example:
    \code
    int n = 10000;
    VectorXd x(n), b(n);
    SparseMatrix<double> A(n,n);
    // fill A and b
    ConjugateGradient<SparseMatrix<double>, Lower|Upper> cg;
    cg.compute(A);
    x = cg.solve(b);
    std::cout << "#iterations:     " << cg.iterations() << std::endl;
    std::cout << "estimated error: " << cg.error()      << std::endl;
    // update b, and solve again
    x = cg.solve(b);
    \endcode
  * 
  * By default the iterations start with x=0 as an initial guess of the solution.
  * One can control the start using the solveWithGuess() method.
  * 
  * ConjugateGradient can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
  *
  * \sa class LeastSquaresConjugateGradient, class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
  */
template <typename _MatrixType, int _UpLo, typename _Preconditioner>
class ConjugateGradient : public IterativeSolverBase<ConjugateGradient<_MatrixType, _UpLo, _Preconditioner>>
{
    typedef IterativeSolverBase<ConjugateGradient> Base;
    using Base::m_error;
    using Base::m_info;
    using Base::m_isInitialized;
    using Base::m_iterations;
    using Base::matrix;

public:
    typedef _MatrixType MatrixType;
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    typedef _Preconditioner Preconditioner;

    enum
    {
        UpLo = _UpLo
    };

public:
    /** Default constructor. */
    ConjugateGradient() : Base() {}

    /** Initialize the solver with matrix \a A for further \c Ax=b solving.
    * 
    * This constructor is a shortcut for the default constructor followed
    * by a call to compute().
    * 
    * \warning this class stores a reference to the matrix A as well as some
    * precomputed values that depend on it. Therefore, if \a A is changed
    * this class becomes invalid. Call compute() to update it with the new
    * matrix A, or modify a copy of A.
    */
    template <typename MatrixDerived> explicit ConjugateGradient(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {}

    ~ConjugateGradient() {}

    /** \internal */
    template <typename Rhs, typename Dest> void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const
    {
        typedef typename Base::MatrixWrapper MatrixWrapper;
        typedef typename Base::ActualMatrixType ActualMatrixType;
        enum
        {
            TransposeInput = (!MatrixWrapper::MatrixFree) && (UpLo == (Lower | Upper)) && (!MatrixType::IsRowMajor) && (!NumTraits<Scalar>::IsComplex)
        };
        typedef typename internal::conditional<TransposeInput, Transpose<const ActualMatrixType>, ActualMatrixType const&>::type RowMajorWrapper;
        EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(MatrixWrapper::MatrixFree, UpLo == (Lower | Upper)),
                            MATRIX_FREE_CONJUGATE_GRADIENT_IS_COMPATIBLE_WITH_UPPER_UNION_LOWER_MODE_ONLY);
        typedef typename internal::conditional<UpLo == (Lower | Upper),
                                               RowMajorWrapper,
                                               typename MatrixWrapper::template ConstSelfAdjointViewReturnType<UpLo>::Type>::type SelfAdjointWrapper;

        m_iterations = Base::maxIterations();
        m_error = Base::m_tolerance;

        RowMajorWrapper row_mat(matrix());
        internal::conjugate_gradient(SelfAdjointWrapper(row_mat), b, x, Base::m_preconditioner, m_iterations, m_error);
        m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
    }

protected:
};

}  // end namespace Eigen

#endif  // EIGEN_CONJUGATE_GRADIENT_H
